Implementation of the Extended Conjugate Gradient Method for the Two- Dimensional Energized Wave Equation

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1 Lonardo Elcronc Jornal of raccs and Tchnolos ISSN Iss 9 Jl-Dcmbr 006 p. -4 Implmnaon of h Endd Cona Gradn Mhod for h Two- Dmnsonal Enrd Wav Eqaon Vcor Onoma WAZIRI * Snda Ass REJU Mahmacs/Compr Scnc Dparmn Fdral Unvrs of Tchnolo Mnna 9000 Nr Sa Nra; * corrspondn ahor Naonal Opn Unvrs of Nra Vcora Island Laos Nra dronomawar@ahoo.com sr@no.d.n Absrac Th papr comps h opmal conrol and sa of h wo-dmnsonal Enrd wav qaon sn h Endd Cona radn Mhod ECGM. Ths pc of work has o do wh all h val compaonal lmns as drvd n h mplmnaon of h ECGM alorhm on h wo-dmnsonal Enrd Wav qaon n War and War & R LEJT & LJS Isss [7-9]. Wh hs rcalls proram cods wr drvd whch av varos nmrcal opmal conrols and sas. Ths opmal conrols and sas wr consdrd a varos pons n a plan srfac. Kwords Endd Cona Gradn Mhod ECGM Cona Gradn Mhod CGM Conrol opraor and pnal cos hp://lp.acadmcdrc.or

2 Implmnaon of h Endd Cona Gradn Mhod for h Two-dmnsonal Enrd Wav qaon Vcor Onoma WAZIRI Snda Ass REJU 4 Inrodcon Th ECGM mplmnaon alorhm for h wo-dmnsonal nrd wav problm s a modfd parn of h CGM alorhm b Hsns and Sfl 5. Th opmal solons of h nrd wav qaon ar obanabl b sbsn h conrol opraor from War al. 7 h analcal solon from War al. 8 and h pnal fnconal lmns from War al. 9 no h CGM mplmnaon framworks modls. Th mplmnaon of h ECGM alorhm In hs opmaon compaonal problm for h wo-dmnsonal nrd qaon h dals ar n ln wh CGM framwork alorhm dvlopd b Hsns al. bd. Snc hs s a compr basd alorhm h nalaon s basd on h dfal of h compr. Th oln dals for h mplmnaon rns sqnall n hs ordr:. α α Eqaons and clarl dfn h mplas for h consrcon of h opmal sa and conrol rspcvl. Th s ' α apparn ar h sp lnhs ha spd p h ra of convrnc whl h p and p dno h cona drcons for h sa and conrols rspcvl. Th sp lnhs for h sa and conrol ar nql dfnd n hs ordrs: B α B α 4 Th pnald dscn drcon sp lnhs ar smlarl nql dfnd as: β 5 β 6

3 Lonardo Elcronc Jornal of raccs and Tchnolos ISSN Iss 9 Jl-Dcmbr 006 p Th or smpl smbol h pnald dscn drcon s wh rspc o h conrol and sa. Th radns of h sa and conrol for h h rms ar smbolcall vn as: B α 7 B α 8 Th h rav dscn alorhm for h sa and conrol ar: β 9 β 0 To mak ffcv compaonal manplaon of h CGM alorhm as porrad n qaons hroh 0 o or problm w ms drv som of h lmns of h modl. Ths consdr or nconsrand problm for h wo-dmnsonal nrd wav qaon from War 006a rprodcd hrndr for convnnc: ] [ mn{ mn J µ [ µ ]} ddd From now onward h nconsran qaon shall b rfrrd o as J µ smbolcall s radns n h conrol parn ar: ] [ { J µ [ µ ]} ddd Th radn wh rspc o h sa racor: ddd J [ { µ

4 Implmnaon of h Endd Cona Gradn Mhod for h Two-dmnsonal Enrd Wav qaon Vcor Onoma WAZIRI Snda Ass REJU Th pnald dscn drcon p n h conrol and sa ar dfnd as follows: ddd 4 p { [ ] µ [ ddd]} 5 Now h prodc of h conrol opraor and h dscn drcon analcall drvd from h conrol opraor n War al. 7. β 4 p 6 54 β p ar ddd 5 [ µ ] ddd 6 4 ddd µ 0 [ ] ddd 6 Th dscn pnald prodc wh h conrol opraor s 6

5 Lonardo Elcronc Jornal of raccs and Tchnolos ISSN Iss 9 Jl-Dcmbr 006 p. -4 B p µ [ ] 0ddd [ µ 6 6 ] 4µ ddd µ [8 6 ] ddd 7 Th drvaons of 6 and 7 wr sccssfl afr mch albrac manplaons from h conrol opraor n War al. 7. In frhranc of h mplmnaon of h ECGM alorhm basd on h problm a hand h nrd wav qaon hs drvaon as acqrd from War al. 7 and 8 ar worh of no: 0 snπsnπ 0 snπsnπ 0 sn π sn π 0 snπsnπ 8 0 snπsnπ 0 snπsnπ 0 sn π sn π 0snπsnπ 9 0sn π sn π 0 snπsnπ 7

6 Implmnaon of h Endd Cona Gradn Mhod for h Two-dmnsonal Enrd Wav qaon Vcor Onoma WAZIRI Snda Ass REJU 0 snπsnπ 0 snπsnπ 0 0sn sn 0sn sn 4 π π π π 4 4 0snπsnπ 0snπsnπ π [ 0 sn π sn π 0snπsnπ 0snπsnπ ] 0snπsnπ wh 0 h follown nal condons ar no dffcl o oban from qaons 8 hroh o : 0 0 sn π sn π 0 sn π sn π 0 0 sn π sn π 0 sn π sn π 4 0 π [ 0 snπsnπ 0 snπsnπ] 5 8

7 Lonardo Elcronc Jornal of raccs and Tchnolos ISSN Iss 9 Jl-Dcmbr 006 p π [ 0 snπsnπ 0 snπsnπ] 6 Sbsn h drvaons of qaons 8 hroh o no qaons 6 and 7 h prodcs of h dscn drcons and h conrol opraor can b oband. Th pnal fnconal bcas of s rol n h ECGM s hrb rcalld from War al. 9: µ 0 snπsnπ 0 snπsnπ 0 snπsnπ 0 snπsnπ / π { 0 snπsnπ 0sn π sn π 0sn π sn π [ π { 0sn π sn π 0 snπsnπ 0 snπsnπ 0 snπsnπ }] 0sn π sn π } { 0sn π sn π 0sn π sn π 0sn π sn π 9

8 Implmnaon of h Endd Cona Gradn Mhod for h Two-dmnsonal Enrd Wav qaon Vcor Onoma WAZIRI Snda Ass REJU 0sn π sn π } 0sn π sn π π π 0sn π sn π 0sn sn 7 0 snπsnπ Havn drvd all h ncssar lmns for h ECGM mplmnaon alorhm; proram cods ar wrn afr all h ncssar sbsonal procsss no h CGM alorhmc parn. Th opmal conrol and sa ops W mak h follown obsrvaons n smmard ablar form ndr varos obsrvaons of h nal amplds and vloc a dsnc plan profls n spposdl absracv spaal plans wh olranc b h applcaon of proram cods. Tabls and rspcvl v h nral smmar of h opmal conrols and sas. Each opmal sa or conrol s a mnmm val from lar daa of nmrcal ras. Th opmal conrols a dffrn chosn plan profls ar: Tabl. Th opmal conrol ops rofl Opmal conrol Th opmal sas spaal plan vals a varos plan profls ar vn n abl : 40

9 Lonardo Elcronc Jornal of raccs and Tchnolos ISSN Iss 9 Jl-Dcmbr 006 p. -4 Tabl. Th opmal conrol ops n-sraa Opmal Sa Conclson W obsrv ha as h sraa profls n spac ncras from n and n 0 h opmal sa vals ar rlavl sabl wh s a nlbl local rror. Bwn n 0 and n 40 h opmal sa and conrol manan sabl a n 50 hr s a slh mp n vals for h sa b a dcras n h conrol. Afr hs pons h opmal sa and conrol vals a n 60 and n 80 rman sabl. Rfrncs [] Hsns M. R. Sfl E. Mhod of cona Gradn Mhod for solvn lnar ssms Jornal of Rsarch of h Naonal Bra of Sandards; p [] Ibba M. A. Onman. On Conrol Opraor and som of s Applcaons J. Mah. Anal. and Applc p. -7. [] Omolhn J. O. On conrol of racon Dffson h.d. Thss Unvrs of Ilorn Nra 99. [4] R S. A. Compaonal Opmaon n Mahmacal phscs; h.d. Thss Unvrs of Ilorn Ilorn Nra 995. [5] Hsns M. A. Sfl Mhod of Cona Gradns for solvn lnar ssms. J. Rs NBS

10 Implmnaon of h Endd Cona Gradn Mhod for h Two-dmnsonal Enrd Wav qaon Vcor Onoma WAZIRI Snda Ass REJU [6] War V. O. Opmal Conrol of Enrd Wav qaons sn h Endd Cona Gradn Mhod ECGM h.d. Thss Fdral Unvrs of Tchnolo Mnna Nra 004. [7] War V. O. R S. A. Conrol Opraor for h Two-Dmnsonal Enrd Wav Eqaon Lonardo Jornal of Scnc ISSN 58-0 Iss 9 Jl-Dcmbr 006 p. -4. [8] War V. O. R S. A. Th Analss of h Two-dmnsonal Dffson Eqaon Wh a Sorc Lonardo Jornal of Scnc ISSN Iss 9 Jl-Dcmbr 006 p.4-5. [9] War V. O. R S. A. Th nal Cos Fnconal for h Two-dmnsonal Enrd Wav Eqaon Lonardo Jornal of Scnc ISSN 58-0 Iss 9 Jl- Dcmbr 006 p

The Penalty Cost Functional for the Two-Dimensional Energized Wave Equation

The Penalty Cost Functional for the Two-Dimensional Energized Wave Equation Lonardo Jornal of Scncs ISSN 583-033 Iss 9, Jly-Dcmbr 006 p. 45-5 Th Pnalty Cost Fnctonal for th Two-Dmnsonal Enrgd Wav Eqaton Vctor Onoma WAZIRI, Snday Agsts REJU Mathmatcs/Comptr Scnc dpartmnt, Fdral